Garibaldi

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Garibaldi is a 7-limit (and higher) temperament of the schismatic family. It is an extension of helmholtz temperament beyond the 5-limit but with the same simple chain-of-fifths structure (so that standard notation may be used). As in helmholtz temperament, 5/4 is mapped to the diminished fourth (e.g. C–F♭), and the new mapping specific to garibaldi is that 7/4 is mapped to the doubly-diminished octave (e.g. C–C𝄫). This makes garibaldi a marvel temperament.

Immediate 11-limit extensions include cassandra (41 & 53), mapping 11/8 to +23 fifths, andromeda (29 & 41), mapping 11/8 to −18 fifths, and helenus (53 & 65d), mapping 11/8 to −30 fifths. Garibaldi is most naturally a 2.3.5.7.19 subgroup temperament due to its immediate availability of 19/16 at the minor third (C–E♭). This is sometimes known as garibaldi nestoria.

Garibaldi was named in honor of Eduardo Sábat-Garibaldi, who developed the dinarra, a 53-tone microtonal guitar in the 1/9-schisma tuning.

Interval chain

In the following table, odd harmonics 1–21 are in bold.

# Cents* Approximate ratios
2.3.5.7.19 subgroup 13-limit extension
Cassandra Andromeda Helenus
0 0.00 1/1
1 702.06 3/2
2 204.12 9/8
3 906.18 27/16, 32/19, 42/25 22/13 22/13 22/13
4 408.24 19/15, 24/19, 63/50, 80/63 14/11
5 1110.29 19/10, 36/19, 40/21 21/11
6 612.35 10/7
7 114.41 15/14, 16/15 14/13
8 816.47 8/5 21/13
9 318.53 6/5 40/33
10 1020.59 9/5, 38/21 20/11
11 522.65 19/14, 27/20 15/11
12 24.71 50/49, 57/56, 64/63, 81/80 40/39, 45/44
13 726.77 32/21 20/13
14 228.82 8/7 15/13
15 930.88 12/7 19/11
16 432.94 9/7 14/11
17 1135.00 27/14, 48/25 52/27 64/33 21/11
18 637.06 36/25, 81/56 13/9 16/11, 19/13
19 139.12 27/25 13/12 12/11 14/13
20 841.18 80/49, 81/50 13/8, 44/27 18/11, 64/39 21/13
21 343.24 60/49 11/9, 39/32 16/13, 27/22 40/33
22 1045.30 64/35 11/6 24/13 20/11
23 547.35 48/35 11/8, 26/19 18/13 15/11
24 49.41 36/35 33/32 27/26 40/39, 45/44
25 751.47 54/35 20/13
26 253.53 81/70, 144/125 22/19 15/13
27 955.59 216/125, 256/147 26/15 19/11
28 457.65 64/49 13/10
29 1159.71 96/49 39/20, 88/45 64/33
30 661.77 72/49 22/15 16/11, 19/13
31 163.83 54/49 11/10 12/11
32 865.88 81/49 33/20 18/11, 64/39
33 367.94 216/175 26/21 16/13, 27/22
34 1070.00 324/175 13/7 24/13
35 572.06 243/175 18/13
36 74.12 256/245 22/21 27/26
37 776.18 384/245 11/7
38 278.24 288/245
39 980.30 432/245
40 482.36 324/245
41 1184.41 486/245

* In 7-limit CTE tuning

Notation

Using garibaldi can be a challenge because it defies the tradition of tertian harmony in chain-of-fifths notation. The just major triad on C is C–Fb–G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.

Cassandra nomenclature for
selected intervals
Ratio Nominal Example
3/2 Perfect fifth C–G
5/4 Downmajor third C–vE
7/4 Downminor seventh C–vBb
11/8 Double-up fourth C–^^F
13/8 Double-up minor sixth C–^^Ab
19/16 Minor third C–Eb
Andromeda nomenclature for selected intervals
Ratio Nominal Example
11/8 Down-diminished fifth
Double-down augmented fourth 		C–vGb
C–vvF#
13/8 Double downmajor sixth C–vvA
Helenus nomenclature for selected intervals
Ratio Nominal Example
11/8 Double-down diminished fifth
Triple-down augmented fourth 		C–vvGb
C–v3F#
13/8 Triple-down major sixth C–v3A

Chords and harmony

Traditional tertian harmony is effective. The default triads on the Pythagorean spine are undevicesimal in quality:

  • 1–19/15–3/2 (C–E–G)
  • 1–19/16–3/2 (C–Eb–G)

Note that the major third also represents 24/19, and the minor third, 13/11. These chords are typically associated with a sort of coldness and metalness, like those in 12edo if not more so.

If a warm, sweet, laid-back sound is desired, the thirds can be inflected inwards by a comma to yield

  • 1–5/4–3/2 (C–vE–G)
  • 1–6/5–3/2 (C–^Eb–G)

Contrarily, for a more sour and active sound, they can be inflected outwards by a comma to yield

  • 1–9/7–3/2 (C–^E-G)
  • 1–7/6–3/2 (C–vEb-G)

Scales

Tunings

Tuning spectra

Garibaldi

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
7\12 700.0000 Lower bound of 9-odd-limit,
2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone
19/16 700.8290 1/3 undevicesimal schisma
19/12 701.1105 1/4 undevicesimal schisma
38\65 701.5385
15/8 701.676 1/7 schisma
5/4 701.711 1/8 schisma
25/24 701.7252 2/17 schisma
5/3 701.738 5-odd-limit minimax, 1/9 schisma
9/5 701.760 1/10 schisma
81/80 701.7922 1/12 schisma
31\53 701.8868
3/2 701.9550 Pythagorean tuning
36/35 702.0321
9/7 702.193 9-odd-limit minimax, 1/16 septimal schisma
7/6 702.209 7-odd-limit minimax, 1/15 septimal schisma
49/48 702.2174 2/29 septimal schisma
7/4 702.227 1/14 septimal schisma
19/10 702.2399
21/16 702.2476 1/13 septimal schisma
64/63 702.2720 1/12 septimal schisma
19/15 702.3111
24\41 702.4390
19/14 702.6079
21/19 702.6732
15/14 702.778
7/5 702.915
21/20 703.1066
17\29 703.4483 Upper bound of 9-odd-limit,
2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone
13/11 703.597

Cassandra

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
7\12 700.0000 Lower bound of 9-odd-limit diamond monotone
19/16 700.8290 1/3 undevicesimal schisma
19/12 701.1105 1/4 undevicesimal schisma
38\65 701.5385
15/8 701.676 1/7 schisma
5/4 701.711 1/8 schisma
25/24 701.7252 2/17 schisma
[0 -10 17⟩ 701.728 5-odd-limit least squares
5/3 701.738 5-odd-limit minimax, 1/9 schisma
9/5 701.760 1/10 schisma
81/80 701.7922 1/12 schisma
19/13 701.8702
31\53 701.8868 Lower bound of 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
15/13 701.9355
13/10 701.9362
3/2 701.9550 Pythagorean tuning
13/8 702.026
13/12 702.030
36/35 702.0321
13/9 702.034
19/11 702.0694
11/10 702.097
15/11 702.102
13/7 702.109 13- and 15-odd-limit minimax
[0 -95 -137 -129 167 143⟩ 702.112 15-odd-limit least squares
21/13 702.1135
[0 -27 7 17⟩ 702.114 9-odd-limit least squares
[0 -38 -80 -122 137 116⟩ 702.128 13-odd-limit least squares
[0 -25 11 35⟩ 702.140 7-odd-limit least squares
[0 17 -52 -88 134⟩ 702.183 11-odd-limit least squares
9/7 702.193 9- and 11-odd-limit minimax, 1/16 septimal schisma
7/6 702.209 7-odd-limit minimax, 1/15 septimal schisma
49/48 702.2174 2/29 septimal schisma
7/4 702.227 1/14 septimal schisma
11/7 702.230
11/8 702.231
21/11 702.2371
19/10 702.2399
11/6 702.244
21/16 702.2476 1/13 septimal schisma
11/9 702.258
64/63 702.2720 1/12 septimal schisma
19/15 702.3111
24\41 702.4390 Upper bound of 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
19/14 702.6079
21/19 702.6732
15/14 702.778
7/5 702.915
21/20 703.1066
17\29 703.4483 Upper bound of 9-odd-limit diamond monotone
13/11 703.597

Andromeda

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
7\12 700.0000 Lower bound of 9- and 11-odd-limit diamond monotone
19/16 700.8290 1/3 undevicesimal schisma
19/12 701.1105 1/4 undevicesimal schisma
38\65 701.5385
15/8 701.676 1/7 schisma
5/4 701.711 1/8 schisma
25/24 701.7252 2/17 schisma
5/3 701.738 5-odd-limit minimax, 1/9 schisma
9/5 701.760 1/10 schisma
81/80 701.7922 1/12 schisma
31\53 701.8868
3/2 701.9550 Pythagorean tuning
36/35 702.0321
9/7 702.193 9-odd-limit minimax, 1/16 septimal schisma
7/6 702.209 7-odd-limit minimax, 1/15 septimal schisma
49/48 702.2174 2/29 septimal schisma
7/4 702.227 1/14 septimal schisma
21/16 702.2476 1/13 septimal schisma
64/63 702.2720 1/12 septimal schisma
19/15 702.3111
24\41 702.4390 Lower bound of 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
19/14 702.6079
11/9 702.630 11-odd-limit minimax
11/6 702.665
21/19 702.6732
11/8 702.705
13/9 702.756 13- and 15-odd-limit minimax
15/14 702.778
13/12 702.792
13/8 702.832
7/5 702.915
19/11 703.0797
21/20 703.1066
19/13 703.1659
15/11 703.359
15/13 703.410
17\29 703.4483 Upper bound of 9-, 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
11/10 703.500
13/10 703.522
13/11 703.597
21/13 701.7817
19/10 702.2399
21/11 703.8926
13/7 704.043
11/7 704.377

Helenus

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
7\12 700.0000 Lower bound of 9- and 11-odd-limit diamond monotone
19/16 700.8290 1/3 undevicesimal schisma
11/7 701.094
19/12 701.1105 1/4 undevicesimal schisma
21/11 701.1149
13/7 701.489
21/13 701.5127
38\65 701.5385 Lower bound of 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
11/10 701.591
15/11 701.607
11/8 701.623
11/6 701.633
11/9 701.644 11-, 13-, and 15-odd-limit minimax
15/8 701.676 1/7 schisma
19/11 701.7109
5/4 701.711 1/8 schisma
25/24 701.7252 2/17 schisma
5/3 701.738 5-odd-limit minimax, 1/9 schisma
9/5 701.760 1/10 schisma
81/80 701.7922 1/12 schisma
13/8 701.802
13/12 701.807
13/9 701.811
13/10 701.831
15/13 701.836
31\53 701.8868 Upper bound of 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
19/13 701.8995
3/2 701.9550 Pythagorean tuning
36/35 702.0321
9/7 702.193 9-odd-limit minimax, 1/16 septimal schisma
7/6 702.209 7-odd-limit minimax, 1/15 septimal schisma
49/48 702.2174 2/29 septimal schisma
7/4 702.227 1/14 septimal schisma
19/10 702.2399
21/16 702.2476 1/13 septimal schisma
64/63 702.2720 1/12 septimal schisma
19/15 702.3111
24\41 702.4390
19/14 702.6079
21/19 702.6732
15/14 702.778
7/5 702.915
21/20 703.1066
17\29 703.4483 Upper bound of 9-odd-limit diamond monotone
13/11 703.597

* Besides the octave

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